Carbon constraints & portfolio choice

The economic question

Suppose an investor chooses between two assets with different expected returns and carbon intensities. What changes when the portfolio must stay within a carbon budget?

We start with a deliberately simple model. Our aim is to understand the constraint before introducing risk, transaction costs or a larger investment universe. The numbers below are illustrative inputs for learning.

A simple mathematical model

Let $\mathbf{w} \in \R^2$ denote portfolio weights, $\boldsymbol{\mu}$ expected returns and $\mathbf{c}$ the assets' carbon intensities. The portfolio has a carbon budget $B$.

Expected portfolio return is $\E[R_p] = \boldsymbol{\mu}^{\top}\mathbf{w}$. We choose the weights to maximise that return:

$$\begin{aligned} \max_{\mathbf{w}} \quad & \boldsymbol{\mu}^{\top}\mathbf{w} \\ \text{subject to} \quad & \mathbf{1}^{\top}\mathbf{w} = 1, \\ & \mathbf{c}^{\top}\mathbf{w} \leq B, \\ & \mathbf{w} \geq 0. \end{aligned}$$

The first constraint means all the capital is allocated. The second limits carbon intensity. The final constraint rules out short positions.

For this example, use the following vectors:

$$\boldsymbol{\mu} = \begin{pmatrix} 0.04 \\ 0.08 \end{pmatrix}, \qquad \mathbf{c} = \begin{pmatrix} 20 \\ 100 \end{pmatrix}.$$

Carbon intensity is expressed in tonnes of CO₂ equivalent per million dollars invested. With weights summing to one, the portfolio's intensity is their weighted average.

Explore the carbon budget

Write $x$ for the weight in the higher-carbon asset. The other weight is $1-x$. Our carbon constraint becomes:

$$20(1-x) + 100x \leq B.$$

Because the second asset has the higher expected return, the model allocates as much as this constraint permits to it. For budgets between 20 and 100:

$$x^{\star} = \frac{B-20}{80}.$$

Adjust the budget below. Compare the allocation, the expected return and the position on the figure.

A TWO-ASSET EXAMPLEIllustrative inputs

Portfolio carbon budget

60 tCO₂e / $m

20 · tighter budget100 · looser budget

Expected return (%)

Higher-carbon asset50%

Expected return6.0%

A tighter carbon budget changes the feasible set. Under these particular inputs, it reduces the best feasible expected return. Changing the inputs or the objective can change that conclusion.

From equation to code

The following Python function reproduces the allocation for budgets in the example's range. It raises an error for a budget below the lowest available carbon intensity.

def allocate_portfolio(budget):
    if budget < 20:
        raise ValueError("No feasible long-only portfolio")

    high_carbon_share = min(1.0, (budget - 20) / 80)
    low_carbon_share = 1.0 - high_carbon_share
    expected_return = (
        0.04 * low_carbon_share
        + 0.08 * high_carbon_share
    )

    return low_carbon_share, high_carbon_share, expected_return

allocate_portfolio(60)
# (0.5, 0.5, 0.06)

This computation makes the relationship transparent: each additional unit of carbon budget permits a little more allocation to the second asset, until its weight reaches one.

Work through an exercise

WORK THROUGH IT

A tighter carbon budget

Set the carbon budget to 40. What fraction of the portfolio can be invested in the higher-carbon asset? What expected return follows?

Try the calculation yourself, then compare it with the interactive example.

Reveal the worked solution+

Starting from the carbon constraint:

$$20(1-x) + 100x \leq 40 \quad \Longrightarrow \quad x \leq \frac{20}{80} = 0.25.$$

The return-maximising allocation is 25% in the higher-carbon asset and 75% in the other asset. The expected return is:

$$0.75 \times 4\% + 0.25 \times 8\% = 5\%.$$

Reflect: if both assets had the same expected return, would the constraint still select a unique portfolio?

What this model leaves open

The constraint is only one way to bring a sustainability consideration into a financial model. A richer analysis could ask:

• How do uncertain returns and covariance affect the allocation?
• What changes if carbon intensity itself is uncertain or measured imperfectly?
• Is the intended goal a portfolio metric, a change in real-world emissions, or both?
• How would we test whether the chosen metric represents the question we care about?

These questions connect the mathematics back to economic reasoning. A clear model is useful when its scope and limits are equally clear.

Continue exploring the research perspective or browse the resource collection.